×

zbMATH — the first resource for mathematics

Quasi-interpolation by quadratic piecewise polynomials in three variables. (English) Zbl 1082.65009
The authors describe a new local spline method which uses quadratic piecewise polynomials in three variables. This method is based on trivariate splines of lowest possible degree while appropriate smoothness conditions are satisfied. They prove that the splines yield nearly optimal approximation order while simultaneously its piecewise derivatives provide optimal approximation of the derivatives for smooth functions. Also, they give the constants (relatively low) of the corresponding error bounds. Several numerical examples are given.

MSC:
65D05 Numerical interpolation
65D07 Numerical computation using splines
41A05 Interpolation in approximation theory
41A15 Spline approximation
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alfeld, P., A trivariate \(C^1\) clough – tocher interpolation scheme, Computer aided geometric design, 1, 169-181, (1984) · Zbl 0566.65003
[2] Alfeld, P.; Piper, B.; Schumaker, L.L., An explicit basis for \(C^1\) quartic bivariate splines, SIAM J. numer. anal., 24, 891-911, (1987) · Zbl 0658.65008
[3] Alfeld, P.; Schumaker, L.L.; Sirvent, M., The dimension and existence of local bases for multivariate spline spaces, J. approx. theory, 70, 243-264, (1992) · Zbl 0761.41007
[4] Alfeld, P.; Schumaker, L.L.; Whiteley, W., The generic dimension of the space of \(C^1\) splines of degree \(d \geqslant 8\) on tetrahedral decompositions, SIAM J. numer. anal., 30, 889-920, (1993) · Zbl 0774.41012
[5] Bajaj, C., Data visualization techniques, (1999), Wiley New York
[6] Barthe, L.; Mora, B.; Dodgson, N.; Sabin, M.A., Triquadratic reconstruction for interactive modelling of potential fields, (), 145-153
[7] Beatson, R.K.; Ziegler, Z., Monotonicity preserving surface interpolation, SIAM J. numer. anal., 22, 2, 401-411, (1985) · Zbl 0579.65011
[8] Boehm, W.; Farin, G.; Kahmann, J., A survey of curve and surface methods in CAGD, Computer aided geometric design, 1, 1-60, (1984) · Zbl 0604.65005
[9] de Boor, C., B-form basics, (), 131-148
[10] Chen, M.; Kaufman, A.E.; Yagel, R., Volume graphics, (2000), Springer Berlin · Zbl 0943.68170
[11] Chui, C.K.; He, X., On the location of sample points for interpolation by \(C^1\) quadratic splines, (), 30-43
[12] Chui, C.K., Multivariate splines, Cbms, vol. 54, (1988), SIAM · Zbl 0644.41007
[13] Dagnino, C., Lamberti, P., 2004. Some performances of local bivariate quadratic \(C^1\) quasi-interpolating splines on non-uniform type-2 triangulations. University of Turin, Department of Mathematics. Preprint · Zbl 1065.65012
[14] Davydov, O.; Zeilfelder, F., Scattered data Fitting by direct extension of local polynomials with bivariate splines, Adv. comp. math., 21, 3-4, 223-271, (2004) · Zbl 1065.41017
[15] Farin, G., Triangular bernstein-Bézier patches, Computer aided geometric design, 3, 83-127, (1986)
[16] Grosse, E., Approximation in VLSI simulation, Numer. algorithms, 5, 591-601, (1993) · Zbl 0789.65004
[17] Haber, J.; Zeilfelder, F.; Davydov, O.; Seidel, H.-P., Smooth approximation and rendering of large scattered data sets, (), 341-347, 571
[18] Hangelbroek, T.; Nürnberger, G.; Rössl, C.; Seidel, H.-P.; Zeilfelder, F., Dimension of \(C^1\) splines on type-6 tetrahedral partitions, J. approx. theory., 131, 157-184, (2004) · Zbl 1062.65014
[19] Holliday, D.J.; Nielson, G.M., Progressive volume models for rectilinear data using tetrahedral coons volumes, (), 83-92
[20] Hoschek, J.; Lasser, D., Fundamentals of computer aided geometric design, (1993), A.K. Peters · Zbl 0788.68002
[21] Jeeawock-Zedek, F., Operator norm and error bounds for interpolating quadratic splines on a non-uniform type-2 triangulation of a rectangular domain, Approx. theory appl., 10, 2, 1-16, (1994) · Zbl 0811.41009
[22] Lai, M.-J.; Le Méhauté, A., A new kind of trivariate \(C^1\) spline, Adv. comp. math., 21, 3-4, 273-292, (2004) · Zbl 1068.41016
[23] Marschner, S.; Lobb, R., An evaluation of reconstruction filters for volume rendering, (), 100-107
[24] Meissner, M.; Huang, J.; Bartz, D.; Mueller, K.; Crawfis, R., A practical comparison of popular volume rendering algorithms, (), 81-90
[25] Mora, B.; Jessel, J.-P.; Caubet, R., Accelerating volume rendering with quantized voxels, (), 63-70
[26] Mora, B.; Jessel, J.-P.; Caubet, R., Visualization of isosurfaces with parametric cubes, (), 377-384
[27] Morgan, J., Scott, R., 1977. The dimension of the space of \(C^1\) piecewise polynomials. Unpublished manuscript
[28] Nielson, G.M., Volume modelling, (), 29-50
[29] Nürnberger, G.; Schumaker, L.L.; Zeilfelder, F., Lagrange interpolation by \(C^1\) cubic splines on triangulations of separable quadrangulations, (), 405-424 · Zbl 1021.41003
[30] Nürnberger, G.; Schumaker, L.L.; Zeilfelder, F., Lagrange interpolation by \(C^1\) cubic splines on triangulated quadrangulations, Adv. comp. math., 21, 3-4, 357-380, (2004) · Zbl 1053.41014
[31] Nürnberger, G.; Zeilfelder, F., Developments in bivariate spline interpolation, J. comput. appl. math., 121, 125-152, (2000) · Zbl 0960.41006
[32] Nürnberger, G.; Zeilfelder, F., Local Lagrange interpolation on powell – sabin triangulations and terrain modelling, (), 227-244 · Zbl 0986.65006
[33] Nürnberger, G.; Zeilfelder, F., Lagrange interpolation by bivariate \(C^1\)-splines with optimal approximation order, Adv. comp. math., 21, 3-4, 381-419, (2004) · Zbl 1064.41005
[34] Parker, S.; Shirley, P.; Livnat, Y.; Hansen, C.; Sloan, P.P., Interactive ray tracing for isosurface rendering, (), 233-238
[35] Prautzsch, H.; Boehm, W.; Paluszny, M., Bézier and B-spline techniques, (2002), Springer Berlin · Zbl 1033.65008
[36] Powell, M.J., Piecewise quadratic surface Fitting for contour plotting, (), 253-277
[37] Powell, M.J.; Sabin, M.A., Piecewise quadratic approximation on triangles, ACM trans. math. software, 4, 316-325, (1977) · Zbl 0375.41010
[38] Rössl, C.; Zeilfelder, F.; Nürnberger, G.; Seidel, H.-P., Visualization of volume data with quadratic super splines, (), 393-400
[39] Rössl, C.; Zeilfelder, F.; Nürnberger, G.; Seidel, H.-P., Reconstruction of volume data with quadratic super splines, IEEE trans. vis. comput. graph., 10, 4, 397-409, (2004)
[40] Sablonnière, P., Error bounds for Hermite interpolation by quadratic splines on an α-triangulation, IMA J. numer. anal., 7, 495-508, (1987) · Zbl 0633.41004
[41] Sablonnière, P., On some multivariate quadratic spline quasi-interpolants on bounded domains, (), 262-278 · Zbl 1040.41004
[42] Sablonnière, P., Quadratic spline quasi interpolants on bounded domain of \(\mathbb{R}^d\), \(d = 1, 2, 3\), Spline and radial functions, Rend. sem. mat. univ. Pol. Torino, 61, 3, 229-246, (2003) · Zbl 1121.41008
[43] Sablonnière, P., 2004. Quadratic B-splines on non uniform criss-cross triangulations of bounded rectangular domains in the plane. Preprint
[44] Schumaker, L.L., Bounds on the dimension of spaces of multivariate piecewise polynomials, Rocky mountain J. math., 14, 251-264, (1984) · Zbl 0601.41034
[45] Schumaker, L.L.; Sorokina, T., Quintic spline interpolation on type-4 tetrahedral partitions, Adv. comp. math., 21, 3-4, 421-444, (2004) · Zbl 1053.41015
[46] Schumaker, L.L.; Sorokina, T., A trivariate box macro element, Constr. approx, (2004)
[47] Sorokina, T., 2004. \(C^1\) multivariate macroelements. PhD Thesis, Vanderbilt University, Nashville, USA
[48] Sorokina, T., Worsey, A., 2004. A multivariate Powell-Sabin interpolant. Preprint · Zbl 1154.65009
[49] Theußl, T.; Möller, T.; Hladuvka, J.; Gröller, M., Reconstruction issues in volume visualization, ()
[50] Thévenaz, P.; Unser, M., High-quality isosurface rendering with exact gradients, (), 854-857
[51] Worsey, A.; Farin, G., An n-dimensional clough – tocher interpolant, Constr. approx., 3, 2, 99-110, (1987) · Zbl 0631.41003
[52] Worsey, A.; Piper, B., A trivariate powell – sabin interpolant, J. cagd, 5, 177-186, (1988) · Zbl 0654.65008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.